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Prog. Theor. Phys. Vol. 44 No. 3 (1970) pp. 689-702

[ Full Text PDF : FREE ACCESS (860K) ]

Clebsch-Gordan Formulas of the SU(1, 1) Group

Haruo Ui

Department of Physics, Tohoku University, Sendai

(Received April 28, 1970)

Abstract:

The Clebsch-Gordan formulas for d-function are studied for both nonunitary finite dimensional representation and unitary infinite dimensional representation of the SU(1, 1) group. It is shown that, in addition to the usual Clebsch-Gordan formulas for the above-mentioned two representations, there exists an additional type of Clebsch-Gordan formula connecting the nonunitary and unitary d-functions. The new kind of Clebsch-Gordan coefficient of the SU(1, 1) obtained in our previous paper arises naturally in this formula. The three Clebsch-Gordan formulas are proved to be interrelated with each other by analytic continuation. It is, further, pointed out that all the recurrence formulas of the unitary d-functions –obtainable from the known recurrence relations of Jacobi polynomials– belong actually to this new type of Clebsch-Gordan formula.
Explicit algebraic formulas of d-functions are derived by employing an elementary algebraic method analogous to the standard treatment of angular momentum.


URL : http://ptp.ipap.jp/link?PTP/44/689/
DOI : 10.1143/PTP.44.689

[ Full Text PDF : FREE ACCESS (860K) ] Citation:

References:

  1. L. C. Biedenharn, “Group Theoretical Approaches to Nuclear Spectroscopy”, Chap. V in Lecture on Theoretical Physics, ed. W. E. Britten (Interscience, New York, 1963), Vol. V.
  2. H. Ui, Ann. of Phys. 49 (1968), 69[CrossRef].
  3. For example, J. Strathdee, J. F. Boyce, R. Delbourgo and A. Salam, “Partial Wave Analysis”, Chap. 6, IC/67/9 (Lecture Notes at International Center for Theoretical Physics, Trieste).
  4. V. Bargmann, Ann. Math. 48 (1947), 568.
  5. L. C. Biedenharn, J. Nuyts and N. Straumann, Ann. Inst. Henri Poincare 3 (1965), A13.
    L. C. Biedenharn, Lecture on Theoretical Physics, Cargese Summer School, Corsica, 1965.
  6. Bateman Manuscript Project, Higher Transcendental Functions (McGraw-Hill, New York, 1954), Vol. I, II, III.
  7. W. J. Holman III and L. C. Biedenharn, Ann. of Phys. 39 (1966), 1[CrossRef]; ibid. 47 (1968), 205[CrossRef].
  8. H. Ui, the succeeding paper, Prog. Theor. Phys. 44 (1970), 703[PTP].

Citing Article(s) :

  1. Progress of Theoretical Physics Vol. 69 No. 4 (1983) pp. 1146-1153 :
    A Class of Simple Hamiltonians with Degenerate Ground State. I
    Gyo Takeda and Haruo Ui
  2. Progress of Theoretical Physics Vol. 72 No. 2 (1984) pp. 266-284 :
    Does Accidental Degeneracy Imply a Symmetry Group?
    Haruo Ui and Gyo Takeda

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